Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No: 132.34902
Autor: Erdös, Pál; Shapiro, H.S.; Shields, A.L.
Title: Large and small subspaces of Hilbert space (In English)
Source: Mich. Math. J. 12, 169-178 (1965).
Review: This paper is concerned with the properties of closed subspaces V of the sequential Hilbert space l2 and of L2(0,1). We shall suffice by quoting the following interesting results of this paper which speak for themselves.
Theorem 1. Let V be a closed linear subspace of l2, and let {\rhon} be given with \rhon \geq 0 and sum \rhon2 < oo. If |x(n)| = O(\rhon) for all x in V, then V is finite-dimensional.
Theorem 3. If V is a closed subspace of l2 and V \subset Ip for some 1 \leq p < 2, then V is finite-dimensional.
Theorem 4. If \rhon \geq 0 and sum \rhon2 = oo, then there exists an infinite-dimensional subspace V of I2 such that sum|x(n)| \rho(n) = oo for all x \ne 0 in V. In the case of L2 (0,1) the situation is different.
The authors quote the well-known result from the theory of Fourier series that there exists an infinite-dimensional closed subspace V of L2 (0,1) such that V \subset Lq for all 1 \leq q < oo and in fact satisfies the condition that int \exp{c|f(x)|2}dx < oo for all c > 0 and all f in V. Then it is shown that if \phi is convex, continuous and strictly increasing on [0,oo) with \phi(0) = 0 and \phi(x)e-cx2 > oo as x > oo for all c > 0, then int \phi(|f|) < oo for all f in V implies that V is finite dimensional. Let V be a closed linear subspace of I2. Then there exist elements \lambdan(n = 1,2,...) in V such that (x,\lambdan) = x(n) for all x in V and dim V = sum ||\lambdan||2. This result is used to prove the following theorem. Theorem 9. Let \phi(z) = sum an zn be an inner function. Then sum n|an|2 = dim(\phi H2)\bot. Thus the Dirichlet integral of \phi is finite (and is then an integral multiple of \pi) if and only if \phi is a finite Blaschke product. The paper finishes with the following question: Does H2 contain an infinite dimensional closed subspace. V with |f(z)| = O(1/(1-|z|)1/4)(|z| < 1).
Reviewer: W.A.J.Luxemburg
Classif.: * 46C05 Geometry and topology of inner product spaces
Index Words: functional analysis